Levinson's theorem for scattering on graphs
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Institute for Quantum Computing colloquium talk delivered on Friday June 18th, 2010.
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Levinson's theorem for scattering on graphs
- 1. Levinson’s Theorem for Scattering on Graphs DJ Strouse University of Southern California Andrew M. Childs University of Waterloo
- 2. Why Scatter on Graphs? • NAND Tree problem: • Best classical algorithm: – Randomized – Only needs to evaluate of the leaves Figure from: Farhi E, Goldstone J, Gutmann S. A Quantum Algorithm for the Hamiltonian NAND Tree.
- 3. Why Scatter on Graphs? Can a quantum algorithm do better?
- 4. Why Scatter on Graphs? • Farhi, Goldstone, Gutmann (2007) • Connections to parallel nodes represent input • Prepare a traveling wave packet on the left… • …let it loose… • …if found on the right after fixed time, answer 1 Figure from: Farhi E, Goldstone J, Gutmann S. A Quantum Algorithm for the Hamiltonian NAND Tree.
- 5. Why Scatter on Graphs? Can scattering on graphs offer a quantum speedup on other interesting problems?
- 6. What You Skipped the BBQ For • Goal: Relate (a certain property of) scattering states (called the winding of their phase) to the number of bound states and the size of a graph 1. Crash Course in Graphs 2. Introduction to Quantum Walks (& scattering on graphs) 3. Meet the Eigenstates i. Scattering states (& the winding of their phase) ii. (The many species of) bound states 4. Some Examples: explore relation between winding & bound states 5. A Brief History of Levinson’s Theorem 6. Explain the Title: “Levinson’s Theorem for Scattering on Graphs” 7. A Sketch of the Proof 8. A Briefer Future of Levinson’s Theorem
- 7. Graphs • Vertices + Weighted Edges Adjacency Matrix
- 8. Quantum Walks • Quantum Dynamics: Hilbert Space + Hamiltonian Basis State For Each Vertex Adjacency Matrix
- 9. Scattering on Graphs Basis states on tail: …where “t” is for “tail”
- 10. Meet the Eigenstates Resolve the identity… Scattering states Bound states Diagonalize the Hamiltonian… Represent your favorite state… …and evolve it!
- 11. Scattering States • Incoming wave + reflected and phase-shifted outgoing wave “scattering” = phase shift
- 13. Scattering States • Incoming wave + reflected and phase-shifted outgoing wave “scattering” = phase shift
- 14. Winding of the Phase
- 15. Standard & Alternating Bound States • SBS: exponentially • ABS: same as SBS but decaying amplitude on with alternating sign the tail Exist at discrete κ depending on graph structure
- 16. Confined Bound States • Eigenstates that live entirely on the graph Eigenstate of G with zero amplitude on the attachment point Exist at discrete E depending on graph structure
- 17. Standard & Alternating Half-Bound States • HBS: constant • AHBS: same as HBS with amplitude on the tail alternating sign •Unnormalizable like SS… but obtainable from BS eqns •Energy wedged between SBS/ABS and SS •May or may not exist depending on graph structure
- 18. Scattering & Bound State Field Guide
- 19. Into the Jungle: Bound States & Phase Shifts in the Wild One SBS & One ABS One HBS & One AHBS One SBS, One ABS, & One CBS No BS
- 20. A Brief History of Levinson’s Theorem • Continuum Case: – Levinson (1949) – No CBS Potential on a half-line (modeling spherically symmetric 3D potential) Excerpt from: Dong S-H and Ma Z-Q 2000 Levinson's theorem for the Schrödinger equation in one dimension Int. J. Theor. Phys. 39 469-81
- 21. A Brief History of Levinson’s Theorem • Continuum Case: – Levinson (1949) – No CBS • Discrete Case: – Case & Kac (1972) • Graph = chain with self-loops • No CBS & ignored HBS – Hinton, Klaus, & Shaw (1991) • Included HBS • …but still just chain with self-loops
- 22. The Theorem
- 23. Proof Outline
- 24. Proof Outline
- 25. Proof Outline Analytic Continuation!
- 26. Proof Outline
- 27. Proof Outline
- 28. Proof Outline
- 29. Proof Outline
- 30. Into the Jungle: Bound States & Phase Shifts in the Wild One SBS & One ABS One HBS & One AHBS One SBS, One ABS, & One CBS No BS
- 31. Future Work • What about multiple tails? – Now R is a matrix (called the S-matrix)… – The generalized argument principle is not so elegant… Excerpt from: H. Ammari, H. Kang, and H. Lee, Layer Potential Techniques in Spectral Analysis, Mathematical Surveys and Monographs, Vol. 153, American Mathematical Society, Providence RI, 2009.
- 32. Future Work • What about multiple tails? – Now R is a matrix (called the S-matrix)… – The generalized argument principle is not so elegant… • Possible step towards new quantum algorithms? – Are there interesting problems that can be couched in terms of the number of bound states and vertices of a graph? – What properties of graphs make them nice habitats for the various species of bound states?
Editor's Notes
- Decision problem
- Worth understanding the scattering scene a little better
- Worth understanding the scattering scene a little better
- This talk spans physics and CS, so each of you is absolved from being expected to know anything. (there are no dumb questions)
- In general in scattering picture, directed graphs with weights to & from complex conjugates of one another, so adjacency matrix is Hermitian, but we simplify to the case with real, symmetric weights. Everything we’ll discuss carries over to the more general case.
- If you know a bit of QM, then you know that to talk about things moving around, we need a Hilbert space and a Hamiltonian.We’ll only need notation for “tail” basis states.
- Long graphs aren’t very interesting. We need a tail to speak of scattering.Most general case = multiple tails (universal for quantum computation)We focus on single tail, so “scattering” involves only a phase shift.With tail, adjacency matrix is now infinite!
- SS normalizable & one speciesBS unnormalizable & many speciesSo this is why we even care about theorems relating scattering and bound states
- Calculations on tails fixes energyCalculations on graphs fix amplitudes on graph (just more linear equations)
- Phi(k) is the property of SS that we’re interested in. More specifically, we’re interested in the winding of Phi(k).
- Zero amplitude on entire tail, including attachment point
- Some of you will spot an unfortunate pun on this page.
- Stress main differences between SS & BS:NormalizabilityLikely locationExistence
- Suggests connectionBased on these examples, you may want to take a guess at what the theorem will look like
- Levinson paper is in another language – both mathematically and linguisticallyLevinson in continuum with no CBS and momenta unboundedOther discrete: chains with self-loops & no CBS
- Levinson paper is in another language – both mathematically and linguisticallyLevinson in continuum with no CBS and momenta unboundedOther discrete: chains with self-loops & no CBS
- And just in case you don’t have a photographic memory, here’s the definitions.Did anyone actually guess this form from the examples?Very odd: bound states & scattering states – orthogonal (expect nothing to do with each other) but somehow related
- AC: Can we just venture off the unit circle into the complex plane like this? Yes, because analytic functions are uniquely defined for this extension.AP: behavior of function inside border tells you something about behavior on border (applies to functions that are analytic except at a finite number of points)
- AC: Can we just venture off the unit circle into the complex plane like this? Yes, because analytic functions are uniquely defined for this extension.AP: behavior of function inside border tells you something about behavior on border (applies to functions that are analytic except at a finite number of points)
- AC: Can we just venture off the unit circle into the complex plane like this? Yes, because analytic functions are uniquely defined for this extension.AP: behavior of function inside border tells you something about behavior on border (applies to functions that are analytic except at a finite number of points)
- AC: Can we just venture off the unit circle into the complex plane like this? Yes, because analytic functions are uniquely defined for this extension.AP: behavior of function inside border tells you something about behavior on border (applies to functions that are analytic except at a finite number of points)
- AC: Can we just venture off the unit circle into the complex plane like this? Yes, because analytic functions are uniquely defined for this extension.AP: behavior of function inside border tells you something about behavior on border (applies to functions that are analytic except at a finite number of points)
- AC: Can we just venture off the unit circle into the complex plane like this? Yes, because analytic functions are uniquely defined for this extension.AP: behavior of function inside border tells you something about behavior on border (applies to functions that are analytic except at a finite number of points)
- AC: Can we just venture off the unit circle into the complex plane like this? Yes, because analytic functions are uniquely defined for this extension.AP: behavior of function inside border tells you something about behavior on border (applies to functions that are analytic except at a finite number of points)
- Suggests connectionBased on these examples, you may want to take a guess at what the theorem will look like